Problem: (2004 USAMO – #5) Let . Prove that
Solution: Note that and always have the same sign. Therefore .
by Holder. QED.
Comment: This is probably my favorite USAMO problem just because the solution is so short and simple. And the fact that I love inequalities. But honestly, can you get another USAMO problem (especially a #5) to give such a quick solution?
Practice Problem: (2003 USAMO – #5) Let be positive real numbers. Prove that
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